According to the Conservative party's 1922 Committee, consisting of senior MPs who run Tory leadership elections, nominations for the post will open on Wednesday and close on Thursday.
"We... Recommend that the process of electing a new leader of the Conservative Party should commence next week... and conclude no later than Friday September 2, although an earlier conclusion may be possible," said committee chief Graham Brady.
"I think the view of the party is that both we as Conservatives and the country more generally, really want certainty, we would like some resolution, and we think it would be a good thing to conclude this process as soon as we practicably can," said Brady.
A list of nominees will then be whittled down to two candidates in a series of votes by Tory MPs, with the final two then voted on by Conservative members.
The new party leader and PM candidate is then expected to be formally declared by September 2.
Cameron, 49, who has been Britain's Prime Minister for six years, announced his resignation at the Downing Street doorstep in his first speech following the results of last week's referendum in favour of Britain's exit from the European Union (EU).
Among those expected to enter the Tory leadership race include leading Brexit campaigner Boris Johnson, UK Home Secretary Theresa May, former defence secretary Liam Fox and Work and Pensions Secretary Stephen Crabb.
It has been suggested UK Chancellor George Osborne could back Johnson in return for the promise of a frontline role in the new Cabinet.
There is also speculation that a new prime minister could call a snap general election later this year, or early 2017, in order to win a personal mandate from the British public.
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